Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives

Abstract

Abstract:In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala's derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations.

Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

“…Consider the scalar test equation y = λy. Substitute hλ =h into (11) and rewrite it in the matrix form to get…”

“…Because of the high cost of evaluating the stages in a fully implicit method, [9] reported that many researchers have opted to reduce it to a diagonally implicit method. Prior works along this line were discussed by [10][11][12][13][14].…”

Switching Point Solution of Second-Order Fuzzy Differential Equations Using Differential Transformation Method

“…The series solution is computed for the upper and lower approximation in each subdomain separately. The initial values at the point t = 0.9 are the terminal values of the previous series, which is y1 = 2.1518γ + 2.6452, y1 = 6.9488 − 2.1518γ y2 = 6.9108γ + 19.1886, y2 = 33.0103 − 6.91086γ (26) A similar method is followed for other points, and the results are represented in Table 6. These results show that the solution has no switching point, and the system has D 2 1,1 differentiability in the domain.…”

“…Fuzzy differential equations are solved using many methods, such as differential or fuzzy inclusion [10][11][12], the extension principle and characterization theorem [13][14][15], and artificial neural networks [16]. Numerical methods such as the differential transformation method [17,18], extension of numerical solution methods for ordinary differential equations (ODEs) [19][20][21][22][23][24], interactive derivatives [25], multistep methods [26], and Runge-Kutta (RK) methods [27] are also applied to solving these equations. The metric properties of fuzzy functions are studied in [28,29].…”

The Extended Block Predictor-Block Corrector Method for Computing Fuzzy Differential Equations

“…This inexactitude extends to the requirement of fuzzy differential equations (FDEs) to surmount the problem. This aspect of fuzzy differential equations takes place in numerous studies like scientific discipline, economic science, psychological science, defense mechanism, human ecology and applied sciences [17], [23], [36] and [39].…”